01 · Lesson 04
Partial differential equations
An ordinary differential equation follows change through one independent variable, usually time. A partial differential equation, or PDE, can follow a biological quantity through time and space simultaneously.
Scenario: a chemical signal spreads through tissue
Cells release a signalling molecule near the centre of a tissue. At first, its concentration is high near the source and low farther away. Molecules move randomly, so the concentration spreads outward while degradation may remove some signal.
Let \(u(x,t)\) be concentration at position \(x\) and time \(t\). Unlike \(A(t)\) in the ODE lesson, this state depends on two independent variables.
Partial derivatives
The time derivative
measures how concentration changes with time at a fixed position. The spatial derivative
measures how concentration changes with position at a fixed time. The symbol \(\partial\) indicates that the function has more than one independent variable.
The second spatial derivative, \(\partial^2u/\partial x^2\), measures spatial curvature. It compares the local concentration with neighbouring concentrations and is central to diffusion.
The diffusion equation
A basic one-dimensional diffusion model is
where \(D>0\) is the diffusion coefficient. If distance is measured in millimetres and time in hours, \(D\) has units mm\(^2\) per hour.
Diffusion moves material down concentration gradients. Peaks tend to flatten and the distribution tends to spread. Diffusion redistributes material; it does not create directed movement toward a biological target.
Flux and conservation
Flux describes the amount crossing a unit boundary per unit time. Fick’s law gives
The minus sign means movement is from higher concentration toward lower concentration. Local conservation states that concentration changes when flux enters or leaves:
Substituting Fick’s law produces the diffusion equation when \(D\) is constant.
Initial and boundary conditions
A PDE needs more information than the equation alone. An initial condition specifies the starting spatial profile:
Boundary conditions describe what happens at the edges of the spatial domain.
| Boundary condition | Example | Possible meaning |
|---|---|---|
| Fixed value | \(u(0,t)=c\) | Concentration maintained at an edge |
| No flux | \(\partial u/\partial x=0\) | Reflecting or impermeable boundary |
| Periodic | \(u(0,t)=u(L,t)\) | Opposite edges are treated as connected |
The same PDE can produce different solutions under different initial or boundary conditions.
Reaction–diffusion equations
Biological quantities may also be produced, removed or transformed locally. Adding a reaction term gives
For signal degradation at rate \(k\), \(f(u)=-ku\). For local logistic population growth, \(f(u)=ru(1-u/K)\). Diffusion represents spatial redistribution; \(f(u)\) represents local biological change.
Multiple interacting species produce coupled reaction–diffusion systems. These can describe invasion fronts, spatial epidemics and biological pattern formation, topics developed later in Spatial Mathematical Biology.
Higher-dimensional space
In two or three spatial dimensions, diffusion uses the Laplacian:
In Cartesian coordinates, \(\nabla^2u\) is the sum of the second derivatives in each spatial direction. The domain may represent a rectangular habitat, a tissue image or a three-dimensional organ, but its geometry and boundaries must be declared.
How PDE solutions are represented
A PDE solution is a function of space and time. It may be displayed as:
- spatial profiles at selected times;
- a colour map with position and time on the axes;
- a surface plot;
- an animation of spatial change; or
- summary quantities such as front position or total mass.
Most realistic biological PDEs require numerical approximation. The spatial domain is divided into grid cells or finite elements, and the resulting discrete equations are advanced through time.
Conditions and limitations
- The state, spatial domain, geometry, units and time interval must be defined.
- Initial and boundary conditions are part of the model, not optional numerical details.
- Diffusion assumes movement can be approximated as continuous random spreading.
- A constant diffusion coefficient assumes spatially uniform mobility.
- Numerical accuracy depends on spatial resolution, time-step choice and method.
- Continuum concentration or density may be unsuitable when very small numbers of individuals or molecules dominate behaviour.
What this lesson adds
You can now distinguish ODEs from PDEs, interpret partial derivatives, diffusion and flux, explain why initial and boundary conditions are required, and separate spatial redistribution from local biological reactions.